15.1: The Time between Successive Generations of Nebulae
Contrary to common belief it is sometimes easier to talk in mathematics than to talk in English; this is the reason why many scientific papers contain more mathematics than is either necessary or desirable. Contrary to common belief it is also often less precise to do so. For mathematical symbols have a tendency to conceal the physical meaning that they are intended to represent; they sometimes serve as a substitute for the arduous task of deciding what is and what is not relevant; they tend to limit the inquiry to those physical circumstances that are represented by them and are by their nature unable to reveal others that may have a decisive influence on the conclusion that is being sought. It is true that mathematics cannot lie. But it can mislead.

However, the dangers of over-indulgence in formula spinning are avoided if mathematics is treated, wherever possible, as a language into which thoughts may only be translated after they have first been expressed in the language of words. The use of mathematics in this way is indeed disciplinary, helpful, and sometimes indispensable.

It is disciplinary in the sense in which it is always disciplinary to translate a statement from one language into another; this is the best way of revealing shoddy thinking. It is helpful because some statements are of such a nature that it is almost, if not quite, impossible to express them in the language of words; it is reasonable to ask to be spared the ordeal of attempting to do so. It is indispensable because quantitative conclusions can only be reached with the help of algebraic expressions.

Such considerations demand now that the comparatively light task be undertaken of seeking algebraic symbols for some of the concepts that have been discussed in previous chapters. Only the simplest mathematics is needed at this stage of the inquiry. But the field of cosmology that is being studied here is complicated and so far almost unexplored. Once clarity is achieved concerning its broad outline, the remaining problems will lie wholly within the mathematician's domain.

It is easy to see that the interval between the moment when a cloud of one generation begins to form and the moment when one of the next generation, and in adjoining space, does so must average the time that it takes for the linear dimensions of space to double themselves. This interval can be calculated with the help of Hubble's constant H.

Let D be a given distance in space. The rate at which this increases is

dD / dt = HD,

the solution of which is

D = k e^Ht

Let D = D₀ at the moment when one puts t = 0 and the above expression becomes

D = D₀ e^Ht         ......         (15a)

H is, according to a recent estimate, 185 kilometres per second per megaparsec, which is the same as 5.84 x 10¹⁹ kilometres per year per megaparsec and has the dimension of reciprocal time. One megaparsec is 3.084 x 10¹⁹ kilometres, from which it follows that

H = 1.89 x 10¹⁰ reciprocal years.

When the distance doubles one puts D / D₀ = 2 and obtains ln2 = Ht, from which t = 3.66 x l0⁹ years.

This means that a new nebula begins to form adjacent to a predecessor about every three-and-a-half thousand million years.

This is the time during which the lip of a crater on an astronomical summit goes through the complete evolution of moving outwards, becoming a ridge far from any concentration and flattening sufficiently to attain at its summit the critical gradient at which a new cloud can begin to form.

15.2: The Rate at which an Extragalactic Cloud Grows in Extent
It is necessary to consider next whether the calculated time that it takes for the cloud to grow is of the right order of magnitude, lf the cosmological model based on Symmetrical Impermanence is to resemble actuality, the cloud must acquire a significant mass during a time that is appreciably less than 3.66 x 10⁹ years.

For the sake of simplicity let us assume that the cloud is forming on a pass between two nebulae of equal masses m. Although the cloud forms on a summit and not on a pass, this incorrect assumption will serve to illustrate the method of calculation and I hope that it will not introduce a wrong order of magnitude. Let D_c0 be the distance from the pass to one of the nebulae at the moment when the cloud just begins at the very top of the pass, and let D_cr be the distance at a moment when the fringe of the cloud reaches a distance r from the pass. Let E_c be the critical potential gradient at which the rate of origins just balances the rate of loss from particles falling away down the slope.

We want to find the time that it takes for the fringe of the cloud to reach a certain fraction of the distance to the nearest galaxy. Let this fraction be α , so that r = α D_cr

Consider equations (12e) and (12f) . lf (12f) were correct, E_c would be reached at distance r at the same time as at every other distance; it would be reached for the separation D_co. But the correct equation is (12e) and for this E_c is not reached until the separation has become D_cr. This gives two expressions for E namely:

E_c = -4Gm D_cr r / ( D_cr² - r² )² from (12e) and,

E_c = - 4G m r / D_cr³ from (12f)

When one replaces r by α D_cr and equates these expressions, one obtains

- 4Gm / D_cr² ( 1 - α ² )² = -4GmD_cr / D_c0³ from which,

D_cr = D_c0 ( l - α² )^-2/3         ......         (15b)

From this one can calculate that when r is 1 per cent of D_cr the ratio D_cr / D_c0 is 1.000067, and when r is 5 per cent of D_cr it is 1.00167.

If one puts for D in equation (15a) D_c0 when t = 0, one can write

D_cr = D_c0e^Ht

From this it follows that t is 350,000 years when the cloud extends to I per cent of the distance to the nearest nebula, and 8.8 million years when the cloud extends to 5 per cent.

These figures are over-estimates, and probably considerable ones, for they do not allow for the flattening of the slope that is occasioned by the mass of the cloud itself. When this is taken into consideration one must arrive at shorter, and probably substantially shorter, times, particularly for the 5 per cent distance.

The estimate does not allow, either, for any change in the mass m of the neighbouring galaxies. The conclusion has been reached in the last section that this is growing only slowly when cloud formation begins. It will be shown in Appendix B that it may be dwindling later. For the condition that is being considered it must be near the turning point, and so neglect of change of mass is not likely to introduce a big error.

It has been found herewith that, compared with the three-and-a-half thousand million years that are available for the cloud to acquire its final mass, the time that it needs to become rather extensive, though it is also still very tenuous, is quite short.

The orders of magnitude in the model based on Symmetrical Impermanence seem to fit.

15.3: The Rate of Growth of the Cloud's Domain
As soon as a cloud has become massive enough to have an appreciable gravitational field of its own, a reversal zone occurs just around the astronomical summit. I have previously described this zone as the lip of a crater. There is some incipient cloud on both sides of this. A portion of this cloud occupies the crater and another portion lies on the outer slopes.

As the mass of the cloud increases, this reversal zone moves outwards; the crater grows in depth and diameter. It continues to do so until the reversal zone coincides with the fringe of the cloud. The first stage of growth then ends and the second stage begins.

We thus have to picture two simultaneous movements, both proceeding radially outwards. One is the movement of the fringe of the cloud, the other that of its reversal zone. This begins like the lip of a crater; at a later stage it envelops the whole cloud, and it extends eventually far out into space, as a range of passes and summits.

The reversal zone can form only after the very tenuous incipient cloud has become fairly extensive, and even then it is no more than a small shell right at the centre of the cloud. To extend eventually beyond the cloud it must grow outwards more quickly than the fringe does. Some simple mathematics shows that this happens.

In practice the cloud's shape must be very irregular, but for the sake of simplicity a spherical shape will be assumed here. This simplification may lead to quantities that would need a fairly large correction factor. But it suffices to give an idea of the relative movements of the fringe and the boundary.

Let at any moment the distance from the centre of the cloud to the reversal zone be D₁ and let the distance from the neighbouring galaxy to the same point on the reversal zone be D₂. Let at the same moment the mass of the cloud within its domain be m₁ and the mass within the domain of the neighbouring galaxy be m₂. By the inverse square law

( D₁ / D₂ )² = m₁ / m₂        ......         (15c)

Let the distance from the centre of the cloud to its fringe be r and let the further simplification be introduced that the mass of the cloud is proportional to its volume. One can then write

m₁ = α r³         ......         (15d)

where α depends on the shape and average density of the cloud. Combining these two equations gives

D₁ = D₂ ( α / m₂ ) ^1/2 r^3/2         ......         (15e)

If D₂, α and m₂ have given values, D₁ is seen from this equation to increase more rapidly than r. This is perhaps easier to appreciate if one replaces equation (15e) by

D₁ / r = constant x r^1/2         ......         (15f)

This expression shows that D₁ is smaller than r when r is small and larger than r when r is large. There is one particular value for r when it equals D₁. This is the value for which the fringe of the cloud coincides with the reversal zone.

Too much importance must not be given to the above equations. They do little more than to provide reassurance that in the model based on Symmetrical Impermanence the time must arrive rather soon in the history of an extragalactic cloud when it ceases to grow in extent, when indeed it begins to shrink under its own gravitational field, and when at the same time it grows in mass by capture of hydrogen from without.

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